Mathematics - Area Under Curves Question with Solution | TestHub

Substitute t=tan x. Since 0<t<1, adding adjacent powers proves the recurrence and bounds:$\begin{array}{c}{A}_n={\int }_0^1\frac{t^n}{1+t^2}dt.\\ A_n+A_{n-2}={\int }_0^1{t}^{n-2}dt\\ =\frac{1}{n-1}.\\ A_{n+2}<A_n<A_{n-2}.\\ 2A_n<A_n+A_{n-2}=\frac{1}{n-1},\\ 2A_n>A_n+A_{n+2}=\frac{1}{n+1}.\\ \frac{1}{2n+2}<A_n<\frac{1}{2n-2}.\end{array}$